I'm trying to use the language of strata to organize $K$-types of irreducible smooth representations of $GL(2)$ (and then hopefully prove things). Unfortunately, I'm still new to it, so I might be making some mistakes.

My main reference is the Bushnell-Henniart book. For anyone who doesn't have access to it and wants to "play along at home", this essay contains a distillation of the important ideas.

Let $k$ be a $p$-adic field with ring of integers $\mathfrak o$, $G=GL_2(k)$, $K=GL_2(\mathfrak o)$. The principal congruence subgroup of level $\varpi^N$ (or level $N$, by abuse of notation) is $K_N=1+\varpi^N M_2(\mathfrak o)$ (and $K_0=K$). Let $\psi$ be an additive character on $k$ with conductor $\mathfrak o$ and extend it to $M_2(k)$ by composing with the trace: $\psi_M(x):=\psi({\rm tr}\ x)$. To further abuse notation, we'll suppress the subscript $M$ in $\psi_M$.

We'll say that the level of an irreducible (smooth) representation $\sigma$ of $K$ is the largest $N$ such that $\sigma$ is nontrivial on $K_N$ and trivial on $K_{N+1}$. Assume $\sigma$ has level $N\ge 1$. Since $K_N/K_{N+1}$ is a finite abelian group, $\sigma|_{K_N}$ decomposes into a direct sum of characters. Using the isomorphism $K_N/K_{N+1}\simeq M_2(\mathfrak o/\varpi)$ (given by $x\rightarrow x-1$), these characters can be written in the form $\psi_a(x):=\psi\big(a(x-1)\big)$ for some $a\in M_2(\mathfrak o/\varpi)$. For our purposes, a stratum is the level $N$ and the character $\psi_a$ on $K_N$.

What strata appear in the $K$-types of irreducible representations of $G$?

My possibly-incorrect understanding is that for a representation that can be compactly-induced from $ZK$, the fundamental stratum of the representation will appear in a $K$-type of lowest level. What happens for higher levels? And what happens for representations associated to the other chain order (than $M_2(\mathfrak o)$)?

For facts about $K$-types of irreducible representations, see Casselman's Restriction paper (this version should be more easily available), or Henniart's Appendix to this paper. For the supercuspidal case, I am aware Hansen's paper (though I don't understand it). The subtext of this paragraph is that I know (in principle) the $K$-types I'm interested in, yet I am still unable to transform this into knowledge of the corresponding strata. Whether this is due to me overlooking something simple or to a more serious issue, I do not know.

As an example, let $\chi_1$ and $\chi_2$ be characters of $k^\times$, with $\chi_1$ unramified and $\chi_2$ ramified with conductor $N_0\ge 1$ (so $\chi_2$ is nontrivial on $1+\varpi^{N_0}\mathfrak o$ but trivial on $1+\varpi^{N_0+1}\mathfrak o$). Set $\chi=\chi_1\otimes\chi_2$, and let ${\rm Ind}_P^G\chi$ be the corresponding ramified principal series. Then ${\rm Ind}_P^G\chi$ contains $K$-types $\sigma_N(\chi)$ of level $N\ge N_0$, where $\sigma_N(\chi)$ is the subrepresentation of ${\rm Ind}_{P\cap K}^K(\chi)$ of level $N$ (we aren't distinguishing between $\chi$ and its restriction to $K\cap P$).

Take $g=\bigg(\matrix{a & b\cr c & d}\bigg)\in K_N$, so that $c=u\varpi^N$, with $u\in\mathfrak o^\times$, then (for example) $$g=\bigg(\matrix{adu^{-1}-b\varpi^N&b\cr &d}\bigg)\bigg(\matrix{1&\cr \varpi^N&1}\bigg)\bigg(\matrix{ud^{-1}&\cr &1}\bigg)$$ Thus, for $v\in \sigma_N(\chi)$ and $g=\bigg(\matrix{a & b\cr c & d}\bigg)\in K_N$ $$\sigma_{N,\chi}(g)\cdot v=\chi_2(d)\sigma_{N,\chi}\Bigg(\bigg(\matrix{1&\cr \varpi^N&1}\bigg)\bigg(\matrix{ud^{-1}&\cr &1}\bigg)\Bigg)\cdot v$$ Since $\chi_2$ has conductor $N_0$, there exists $a_2$ such that $\chi_2(d)=\psi\big(a_2(d-1)\big)$. When $d\in 1+\varpi^N$ for $N>N_0$, the character will be trivial. I'm happy with this, but I don't understand how to get the rest of the calculation to work out, though I feel it should be a straight-forward exercise.